146
6 Longitudinal Magnetic Field Effect
the surface magnetic field changes, and it may be expected that flux lines are driven
by the Lorentz force as the critical state model assumes. In this case the critical state
may be reached when the surface magnetic field matches the penetration field and
the flux lines penetrating from the two surfaces meet at the center of the slab. The
observed critical current density was much larger than that in the transverse magnetic
field, however, and was equal to that in the usual longitudinal magnetic field. This
shows that the current is free to flow along the direction that minimizes the loss
energy.
Thus, the effect of flux pinning does not appear in the force-balance, whereas it
actually determines the critical state. The force-balance equation in the force-free
condition represents the characteristic feature of the longitudinal field effects. It is not
an essential equation, however, but is a shadow like a “cast-off skin” of the pinning
effect.
As is shown above, it can be empirically demonstrated that the principle of
minimum energy dissipation is applicable to various examples of flux pinning
phenomena. This principle is also deeply associated with the determination of the
flux bundle size in flux creep
3 [24], as well as the critical state in the transverse
magnetic field and the sharing of pinning energy in the longitudinal magnetic field.
This principle is known to hold for a linear dissipation system. On the other hand,
the flux pinning phenomena belong to nonlinear case. The critical state model holds,
however, in the quasi-static process accompanied by pinning loss. Hence, this principle may be applicable to nonlinear systems so long as the energy dissipation is
small, and further discussion is desirable (see Appendix A.10).
Based on the above discussion on the remaining two points, we can say that the
theoretical framework to explain the electromagnetic phenomena in the longitudinal
magnetic field is completed.
6.6 Comparison with Electromagnetic Phenomena
in the Transverse Magnetic Field
As shown in this chapter, the electromagnetic phenomena in the longitudinal
magnetic field are dramatically different from those in the transverse magnetic field.
In this section the differences between them are compared.
The explanation of each of the peculiar longitudinal magnetic field effects is
listed in Table 6.1. Understanding them is essentially different from understanding
the electromagnetic phenomena in the transverse magnetic field. The existence of
the force-free torque was unknown even in traditional electromagnetism, since the
3 Flux creep: The state in which flux lines are captured by pinning centers is a non-equilibrium
state, and hence, it can happen that flux lines are depinned by thermal agitation. This phenomenon
is flux creep and is remarkable in high-temperature superconductors. A group of flux lines that are
simultaneously depinned is called a flux bundle. The superconducting current density determined
by flux pinning interactions sometimes decreases with time due to flux creep. When the effect of
the flux creep is significant, the critical current density sometimes is reduced to zero.
6 Longitudinal Magnetic Field Effect
the surface magnetic field changes, and it may be expected that flux lines are driven
by the Lorentz force as the critical state model assumes. In this case the critical state
may be reached when the surface magnetic field matches the penetration field and
the flux lines penetrating from the two surfaces meet at the center of the slab. The
observed critical current density was much larger than that in the transverse magnetic
field, however, and was equal to that in the usual longitudinal magnetic field. This
shows that the current is free to flow along the direction that minimizes the loss
energy.
Thus, the effect of flux pinning does not appear in the force-balance, whereas it
actually determines the critical state. The force-balance equation in the force-free
condition represents the characteristic feature of the longitudinal field effects. It is not
an essential equation, however, but is a shadow like a “cast-off skin” of the pinning
effect.
As is shown above, it can be empirically demonstrated that the principle of
minimum energy dissipation is applicable to various examples of flux pinning
phenomena. This principle is also deeply associated with the determination of the
flux bundle size in flux creep
3 [24], as well as the critical state in the transverse
magnetic field and the sharing of pinning energy in the longitudinal magnetic field.
This principle is known to hold for a linear dissipation system. On the other hand,
the flux pinning phenomena belong to nonlinear case. The critical state model holds,
however, in the quasi-static process accompanied by pinning loss. Hence, this principle may be applicable to nonlinear systems so long as the energy dissipation is
small, and further discussion is desirable (see Appendix A.10).
Based on the above discussion on the remaining two points, we can say that the
theoretical framework to explain the electromagnetic phenomena in the longitudinal
magnetic field is completed.
6.6 Comparison with Electromagnetic Phenomena
in the Transverse Magnetic Field
As shown in this chapter, the electromagnetic phenomena in the longitudinal
magnetic field are dramatically different from those in the transverse magnetic field.
In this section the differences between them are compared.
The explanation of each of the peculiar longitudinal magnetic field effects is
listed in Table 6.1. Understanding them is essentially different from understanding
the electromagnetic phenomena in the transverse magnetic field. The existence of
the force-free torque was unknown even in traditional electromagnetism, since the
3 Flux creep: The state in which flux lines are captured by pinning centers is a non-equilibrium
state, and hence, it can happen that flux lines are depinned by thermal agitation. This phenomenon
is flux creep and is remarkable in high-temperature superconductors. A group of flux lines that are
simultaneously depinned is called a flux bundle. The superconducting current density determined
by flux pinning interactions sometimes decreases with time due to flux creep. When the effect of
the flux creep is significant, the critical current density sometimes is reduced to zero.